I've been involved with many volunteer-driven organizations over the years; in fact, working with these types of groups as both a volunteer and as an organizer of volunteers has been a big part of my life, so I do have some feel for this. The best (and most successful) organizations never treat volunteering as a black-box experience. They outreach constantly, always try to find a place for people who indicate a desire to help, never leave a potential volunteer hanging. Volunteering is an act of personal generosity. If an organization treats volunteering as though it was a privilege, then (whether they realize it or not) they are self-limiting their effectiveness.
Back when it was easy to look at the list of members of this forum, I was always surprised by the number of long-time active members who were not moderators. Dozens of them in the top 100 posters alone -- all obviously committed to the forum over a period of many years. Clearly few of them had been asked to help, because few of them would be likely to refuse. So when I hear that the problem is not enough moderators, I know that the real problem is a lack of asking.